I've recently upgraded my computer and got the Intel 3570k as my cpu. After installing everything ( The cooler is Zalman 9500 and thermal paste is Tuniq-TX2 ), I noticed that my second core has much higher temperature than the other cores. I tried reapplying the paste, but it stayed at being relatively higher than the other cores. Is this normal?

It is completely normal. Nothing to worry about. Enjoy your pc and have fun with it. Thanks
Edit: Only other thing would be too much paste or not enough. When you removed the heat sink how does the paste look?

It was all over the CPU, didnt really notice any spots where its more than normal. Also, initially I used the pea sized method, and the second time the vertical line method, and both resulted with same temps as above.

I had one core that was abnormally hotter then the rest(~5-7c), it turns out i touched the cpu sometime during installation very slightly, cleaned and everything was much lower and cooler, now i only have 1 core that is cooler then the others by usually 3c.

During both of my applications, all that might have gone wrong is that after setting down the heatsink, I have moved it to the side a tiny bit in order to be able to screw it down, I don't think that can cause such problems right?